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Impartial games

Combinatorial game theorystudies games in which two players
take turns, each player has complete information, and there is no chance
involved. A game is over when there are no legal moves left, and
typically the winner is determined by who made the last move. In normal play, the last player to move is the winner, and in misère play, the last player to move is the loser.

Misère play has a much more complicated theory than normal play.

A game is impartialif it satisfies the additional
property that the legal moves from each position are the same for both
players. (This excludes, for example, chess, in which one player plays
the white pieces and the other plays the black pieces.)